Another framework used to obtain PN corrections to the dynamics of a binary coalescing system was found in 1971 by Iwasaki [1]}, where he showed that one can obtain a 1PN gravitational potential by evaluating relativistic scattering amplitudes in a quantum field theory of gravity coupled to matter, described by an action made by the Einstein-Hilbert action plus the action of a scalar field minimally coupled to gravity. Later on, treating General Relativity as a low-energy Effective Field Theory of a Quantum theory of Gravity [2]}, [3]}, [4]}, [5]}, it was proved that the loop expansion used in this approach allow one to distinguish between classical and quantum terms appearing in the amplitude, because of a subtle cancellation of \(\hbar \)  factors [6]}, and this method allowed the evaluation of classical and quantum general relativistic corrections to the Newtonian potential [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}. In recent times this particular method has been receiving a renewed interest, since one can make computations using advanced amplitudes techniques originally developed for Yang-Mills theories computations, such as unitarity methods [14]}. These new techniques extremely simplify the evaluation of gravity amplitudes and make such an approach a good candidate for further advance in the formulation of PN models. Moreover, scattering amplitudes are organized in the so-called Post-Minkowskian (PM) perturbative scheme [15]}, [16]}, [17]}, which is a weak-field only expansion where at \(n\) -PM order one considers contributions that scale as \(G_N^n\) . The PM scheme has been lately proposed to compute the two-body potential at 2PM order, first in [18]} using an EFT matching procedure, and later in [19]} using instead a relativistic Lippmann-Schwinger equation.
The evaluation has then been extended at 3PM order [20]}, [21]}, [22]}, [23]}, [24]}, and at 4PM [25]}, [26]}, [27]} using different techniques. This framework can be used to study other relevant physical problems, such as the bending of light under the influence of a heavy object like the sun [28]}, [29]}, [30]}, [31]}, [32]}. In perturbative quantum gravity, one can study both classical and quantum corrections to this process and to different physical observables, like the Saphiro time delay, using Scattering Amplitudes multi-loop techniques. In particular, this kind of analysis can give us unique estimates of quantum gravity corrections, which can provide new insights on the UV completions of gravitational theories, helping in the development of EFTs of modified gravity [33]}.
